CheiRank

Definition

For a given directed network the Google matrix is constructed in the way described in the article Google matrix. The PageRank vector is the eigenvector with the maximal real eigenvalue \lambda=1 . It was introduced in{{Citation P^_i \propto 1/{K^_i}^{\beta^} with \beta^ =1/(\nu^-1) \approx 0.6 where \nu^ \approx 2.7 is the exponent for outgoing links distribution of the WWW. The CheiRank was introduced for the procedure call network of Linux Kernel software in,{{Cite arXiv |doi-access = free

Examples

An example of nodes distribution in the plane of PageRank and CheiRank is shown in Fig.1 for the procedure call network of Linux Kernel software.

The dependence of P, P^* on K, K^* for the network of hyperlink network of Wikipedia English articles is shown in Fig.2 from Zhirov. The distribution of these articles in the plane of PageRank and CheiRank is shown in Fig.3 from Zhirov. The difference between PageRank and CheiRank is clearly seen from the names of Wikipedia articles (2009) with highest rank. At the top of PageRank we have 1.United States, 2.United Kingdom, 3.France while for CheiRank we find 1.Portal:Contents/Outline of knowledge/Geography and places, 2.List of state leaders by year, 3.Portal:Contents/Index/Geography and places. Clearly PageRank selects first articles on a broadly known subject with a large number of ingoing links while CheiRank selects first highly communicative articles with many outgoing links. Since the articles are distributed in 2D they can be ranked in various ways corresponding to projection of 2D set on a line. The horizontal and vertical lines correspond to PageRank and CheiRank, 2DRank combines properties of CheiRank and PageRank as it is discussed in Zhirov. It gives top Wikipedia articles 1.India, 2.Singapore, 3.Pakistan.

The 2D ranking highlights the properties of Wikipedia articles in a new rich and fruitful manner. According to the PageRank the top 100 personalities described in Wikipedia articles have in 5 main category activities: 58 (politics), 10 (religion),17 (arts), 15 (science), 0 (sport) and thus the importance of politicians is strongly overestimated. The CheiRank gives respectively 15, 1, 52, 16, 16 while for 2DRank one finds 24, 5, 62, 7, 2. Such type of 2D ranking can find useful applications for various complex directed networks including the WWW.

CheiRank and PageRank naturally appear for the world trade network, or international trade, where they and linked with export and import flows for a given country respectively.{{Cite journal

Possibilities of development of two-dimensional search engines based on PageRank and CheiRank are considered.{{Cite journal

Simple network example

A simple example of the construction of the Google matrices G and G^* , used for determination of the related PageRank and CheiRank vectors, is given below. The directed network example with 7 nodes is shown in Fig.4. The matrix S , built with the rules described in the article Google matrix, is shown in Fig.5; the related Google matrix is G=\alpha S + (1-\alpha)e e^{T}/N and the PageRank vector is the right eigenvector of G with the unit eigenvalue ( G P=P). In a similar way, to determine the CheiRank eigenvector all directions of links in Fig.4 are inverted, then the matrix S^* is built, according to the same rules applied for the network with inverted link directions, as shown in Fig.6. The related Google matrix is G^=\alpha S^ + (1-\alpha)e e^{T}/N and the CheiRank vector is the right eigenvector of G^* with the unit eigenvalue ( G^* P^=P^). Here \alpha \approx 0.85 is the damping factor taken at its usual value.

References

References

1. Langville, Amy N.. (2006). "[[Google's PageRank and Beyond]]". [[Princeton University Press]].